Rewriting expressions with logarithm and exponent rules

Statement

Rewrite the following expressions in an equivalent form (do not compute them), using the rules of logarithms and exponents:

• (a) $2^{93}\cdot 2^{77}$
• (b) $\left(2^{12}\right)^{13}$
• (c) $\log(123\cdot 321)$
• (d) $\log\!\left(\dfrac{123}{321}\right)$
• (e) $\log\!\left(4^{123}\right)$

Solution

• (a) Product of powers with the same base: $2^{93}\cdot 2^{77}=2^{93+77}=2^{170}$.
• (b) Power of a power: $\left(2^{12}\right)^{13}=2^{12\cdot 13}=2^{156}$.
• (c) Logarithm of a product: $\log(123\cdot 321)=\log 123+\log 321$.
• (d) Logarithm of a quotient: $\log\!\left(\dfrac{123}{321}\right)=\log 123-\log 321$.
• (e) Logarithm of a power: $\log\!\left(4^{123}\right)=123\log 4=246\log 2$.

$\boxed{2^{170};\quad 2^{156};\quad \log 123+\log 321;\quad \log 123-\log 321;\quad 123\log 4}$